This is wrong. The blurred image contains only intensity information, but reversing the convolution in frequency space would require phase information as well. A simple Gaussian blur is not reversible, even in principle.
There is no "phase information" in the spatial domain. "Phase" is literally, where the pixels are on the screen.
Rather, reversing blur of any type is limited (a) by spatial decimation (a.k.a. down sampling, which is performed in the article), and (b) by noise/quantization floor, below which high frequency content has been pushed.
The input of the DFT is real, but the output is complex. Filtering in the Fourier domain means that the DFT of the image is multiplied with that of the filter. The resulting complex array is than converted into an output image by taking the magnitude. This destroys the phase information and makes the operation irreversible.
Another point of view is that there are infinitely many images that will produce the same result after blurring. Obviously, this makes the operation irreversible.
> The resulting complex array is than converted into an output image by taking the magnitude.
No, it's emphatically not. Perhaps you are thinking of displaying a spectrogram.
To produce an image from frequency-domain data, inverse DFT must be applied. Since (as @nyanpasu64 points out), the DFT of a real-valued image or kernel is conjugate-symmetric (and vice-versa), the result is again real-valued without loss of information. The phase information is not lost. If it were, the image would be a jumbled mess.
(Not that DFT+inverse DFT is necessary for Gaussian blur anyway -- you simply convolve with a truncated Gaussian kernel.)
> Another point of view is that there are infinitely many images that will produce the same result after blurring.
No, this is not true. I don't know why you think it is. This is only true of a brick wall filter, which Gaussian filter is not [1].
The SNR of high-spatial-frequency components is reduced for sure, which can lead to irrevocable information loss. But this is nothing to do with phase.
Both an image and a convolution have a conjugate-symmetric DFT, and the phase of each complex bin encodes spatial information, and there's no "taking the magnitude" involved anywhere when turning a spectrum back into an image (only complex cancellation).
